Calculo vectorial

As we can see from Figure 4.1.1, over a typical subinterval [ti, ti+1] the distance ¢si traveled
along the curve is approximately
q
¢xi
2 +¢yi
2, by thePythagorean Theorem. Thus, if the
subinterval is small enough then the work done in moving the object along that piece of the
curve is approximately
Force × Distance ≈ f (xi∗, yi∗
)
q
¢xi
2 +¢yi
2 , (4.1)
where (xi∗
, yi∗
) = (x(ti∗), y(ti∗)) for some ti∗ in [ti, ti+1], and so
W ≈
nX−1
i=0
f (xi∗
, yi∗
)
q
¢xi
2 +¢yi
2(4.2)
is approximately the total amount of work done over the entire curve. But since
q
¢xi
2 +¢yi
2 =
sµ
¢xi
¢ti
¶2
+
µ
¢yi
¢ti
¶2
¢ti ,
where ¢ti =ti+1 −ti, then
W ≈
nX−1
i=0
f (xi∗
, yi∗
)
sµ
¢xi
¢ti
¶2
+
µ
¢yi
¢ti
¶2
¢ti . (4.3)
Taking the limit of that sum as the length of the largest subintervalgoes to 0, the sum over
all subintervals becomes the integral from t = a to t = b,
¢xi
¢ti
and
¢yi
¢ti
become x′(t) and y′(t),
respectively, and f (xi∗
, yi∗) becomes f (x(t), y(t)), so that
W =
Zb
a
f (x(t), y(t))
q
x′(t)2 + y′(t)2 dt . (4.4)
The integral on the right side of the above equation gives us our idea ofhow to define,
for any real-valued function f (x, y), the integral of f (x, y) along the curve C, called a line
integral:
Definition 4.1. For a real-valued functionf (x, y) and a curve C in R2, parametrized by
x = x(t), y = y(t), a ≤ t ≤ b, the line integral of f (x, y) along C with respect to arc length
s is Z
C
f (x, y)ds =Zb
a
f (x(t), y(t))
q
x′(t)2 + y′(t)2 dt . (4.5)
The symbol ds is the differential of the arc length function
s = s(t) =
Zt
a
q
x′(u)2 + y′(u)2 du , (4.6)